Christophe Croux

In regression analysis, the presence of outliers in the dataset can strongly distort the classical least-squares estimator and lead to unreliable results. To deal with this, several robust-to-outliers methods have been proposed in the statistical literature. In Stata, some of these methods are available through the rreg and qreg commands. Unfortunately, these methods resist only some specific types of outliers and turn out to be ineffective under alternative scenarios. In this article, we present more effective robust estimators that we implemented in Stata. We also present a graphical tool that recognizes the type of detected outliers.

A well known problem for precise positioning in real environments is the presence of outliers in the measurement sample. Its importance is even bigger in ultrasound based systems since this technology needs a direct line of sight between emitters and receivers. Standard techniques for outlier detection in range based systems do not usually employ robust algorithms, failing when multiple outliers are present. The direct application of standard robust regression algorithms fails in static positioning (where only the current measurement sample is considered) in real ultrasound based systems mainly due to the limited number of measurements and the geometry effects. This paper presents a new robust algorithm, called RoPEUS, based on MM estimation, that follows a typical two-step strategy: 1) a high breakdown point algorithm to obtain a clean sample, and 2) a refinement algorithm to increase the accuracy of the solution. The main modifications proposed to the standard MM robust algorithm are a built in check of partial solutions in the first step (rejecting bad geometries) and the off-line calculation of the scale of the measurements. The algorithm is tested with real samples obtained with the 3D-LOCUS ultrasound localization system in an ideal environment without obstacles. These measurements are corrupted with typical outlying patterns to numerically evaluate the algorithm performance with respect to the standard parity space algorithm. The algorithm proves to be robust under single or multiple outliers, providing similar accuracy figures in all cases.